What are some of the problems that arise from introducing too many axioms?

Question

From Mathigon:

If there are too many axioms, you can prove almost anything, and mathematics would also not be interesting.

I don't understand this. Why does introducing more axioms cause a problem?

Answer 1

I'm not sure how rigorous that statement is meant to be. But if one of your axioms contradicts another, every proposition is provable. That's a big problem.

It's not as if mathematicians spend a lot of time debating whether there are too many axioms or too few. But there are some notable exceptions. For thousands of years mathematicians tried to prove that Euclid's fifth postulate followed from the others. As it turned out, eliminating that axiom revealed beautiful noneuclidean geometries, totally consistent with the other four postulates. So adopting a framework with fewer axioms led to many more theorems.

Also, as the article points out, the Axiom of Choice is a famous axiom of set theory. It seems “obvious” on its face, and there seems to be no reason to adopt it as an axiom. And yet...it gives us such counter-intuitive theorems as the Banach-Tarski paradox. But we can't have one without the other.